document.write( "Question 189832: Factor completely\r
\n" ); document.write( "\n" ); document.write( "30x^2 + 4x – 2 \n" ); document.write( "

Algebra.Com's Answer #142433 by jim_thompson5910(35256) $\"\"$ $\"About$

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Looking at the expression $\"30x%5E2%2B4x-2\"$ , we can see that the first coefficient is $\"30\"$ , the second coefficient is $\"4\"$ , and the last term is $\"-2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"30\"$ by the last term $\"-2\"$ to get $\"%2830%29%28-2%29=-60\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-60\"$ (the previous product) and add to the second coefficient $\"4\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-60\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-60\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,10,12,15,20,30,60\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-60\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-60)
\n" ); document.write( "2*(-30)
\n" ); document.write( "3*(-20)
\n" ); document.write( "4*(-15)
\n" ); document.write( "5*(-12)
\n" ); document.write( "6*(-10)
\n" ); document.write( "(-1)*(60)
\n" ); document.write( "(-2)*(30)
\n" ); document.write( "(-3)*(20)
\n" ); document.write( "(-4)*(15)
\n" ); document.write( "(-5)*(12)
\n" ); document.write( "(-6)*(10)\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"4\"$ :\r
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First Number	Second Number	Sum
1	-60	1+(-60)=-59
2	-30	2+(-30)=-28
3	-20	3+(-20)=-17
4	-15	4+(-15)=-11
5	-12	5+(-12)=-7
6	-10	6+(-10)=-4
-1	60	-1+60=59
-2	30	-2+30=28
-3	20	-3+20=17
-4	15	-4+15=11
-5	12	-5+12=7
-6	10	-6+10=4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-6\"$ and $\"10\"$ add to $\"4\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-6\"$ and $\"10\"$ both multiply to $\"-60\"$ and add to $\"4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"4x\"$ with $\"-6x%2B10x\"$ . Remember, $\"-6\"$ and $\"10\"$ add to $\"4\"$ . So this shows us that $\"-6x%2B10x=4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"30x%5E2%2Bhighlight%28-6x%2B10x%29-2\"$ Replace the second term $\"4x\"$ with $\"-6x%2B10x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2830x%5E2-6x%29%2B%2810x-2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%285x-1%29%2B%2810x-2%29\"$ Factor out the GCF $\"6x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%285x-1%29%2B2%285x-1%29\"$ Factor out $\"2\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%2B2%29%285x-1%29\"$ Combine like terms. Or factor out the common term $\"5x-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"30x%5E2%2B4x-2\"$ factors to $\"%286x%2B2%29%285x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%286x%2B2%29%285x-1%29\"$ to get $\"30x%5E2%2B4x-2\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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